| L(s) = 1 | + (−1.68 − 1.08i)2-s + (−0.426 − 2.96i)3-s + (1.66 + 3.63i)4-s + (4.71 + 1.38i)5-s + (−2.49 + 5.45i)6-s + (3.70 − 4.27i)7-s + (1.13 − 7.91i)8-s + (−8.63 + 2.53i)9-s + (−6.43 − 7.42i)10-s + (37.6 − 24.2i)11-s + (10.0 − 6.48i)12-s + (34.4 + 39.7i)13-s + (−10.8 + 3.18i)14-s + (2.09 − 14.5i)15-s + (−10.4 + 12.0i)16-s + (3.93 − 8.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (−0.0821 − 0.571i)3-s + (0.207 + 0.454i)4-s + (0.421 + 0.123i)5-s + (−0.169 + 0.371i)6-s + (0.200 − 0.230i)7-s + (0.0503 − 0.349i)8-s + (−0.319 + 0.0939i)9-s + (−0.203 − 0.234i)10-s + (1.03 − 0.663i)11-s + (0.242 − 0.156i)12-s + (0.735 + 0.848i)13-s + (−0.207 + 0.0608i)14-s + (0.0360 − 0.251i)15-s + (−0.163 + 0.188i)16-s + (0.0562 − 0.123i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0120 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0120 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.916013 - 0.927091i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.916013 - 0.927091i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.68 + 1.08i)T \) |
| 3 | \( 1 + (0.426 + 2.96i)T \) |
| 23 | \( 1 + (-36.5 + 104. i)T \) |
| good | 5 | \( 1 + (-4.71 - 1.38i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (-3.70 + 4.27i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (-37.6 + 24.2i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-34.4 - 39.7i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-3.93 + 8.62i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (65.4 + 143. i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-122. + 269. i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (31.5 - 219. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (34.1 - 10.0i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-56.3 - 16.5i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (46.8 + 325. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 - 438.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (100. - 115. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-340. - 393. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (45.5 - 316. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (35.3 + 22.6i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (93.6 + 60.1i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (160. + 350. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (-877. - 1.01e3i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (162. - 47.8i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-87.5 - 609. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (485. + 142. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.23789723800984311598247387441, −11.38291348408271707962955060080, −10.56205960140408868944241506669, −9.134373641299961551214705150487, −8.502463511634880660367190104435, −6.97150845875525846587999358972, −6.21794026227117705553514091568, −4.22260487996969153306466616191, −2.42464777672107233870051914930, −0.890850330080443164001009547420,
1.57882413153762669917908091864, 3.77427231997011669510451596122, 5.39609309023896020503718009837, 6.33514187102803487833770435802, 7.82972687679222009603662733364, 8.891477505678556591198314390467, 9.785664182895672278435207369389, 10.65689398111328483486645349669, 11.78568217023667703198714433564, 12.94588152229045821913753062717
